Joe,
You and I are a lot alike... we both try to over complicate tasks, and in so doing we risk getting overwhelmed.

Instead of trying to hear ALL chords in ALL songs (too complicated) I suggest focusing on the nashville system, in which you learn where the I-IV-V etc chords are located for one key. You should actually be able to do this pretty quickly through repetition.

Once you know the pattern, take Herb's advice and get the key from the last chord in the song. Maybe not 100% accurate, but hey, we're trying to simplify a complex task here.

Once you know the pattern and the key, just use a capo to put you in the right key, and play the one pattern that you have learned. With your experience, you should be able to get from where you are now to reasonable proficiency in a pretty short period.